pubs.acs.org/JPCL Letter
Compact Bases for Vibronic Coupling in Spectral Simulations: The Photoelectron Spectrum of Cyclopentoxide in the Full 39 Internal Modes Yifan Shen* and David R. Yarkony*
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ABSTRACT: We report an algorithm to automatically generate compact multimode vibrational bases for the Köppel−Domcke−Cederbaum (KDC) vibronic coupling wave function used in spectral simulations of moderate-sized molecules. As a full quantum method, the size of the vibronic expansion grows exponentially with respect to the number of vibrational modes, necessitating compact bases for moderate-sized systems. The problem of generating such a basis consists of two parts: one is the choice of vibrational normal modes, and the other is the number of phonons allowed in each mode. A previously developed final-state-biased technique addresses the former part, and this work focuses on the latter part: proposing an algorithm for generating an optimal phonon distribution. By virtue of this phonon distribution, compact and affordable bases can be automatically generated for systems with on the order of 15 atoms. Our algorithm is applied to determine the nonadiabatic photoelectron spectrum of cyclopentoxide in the full 39 internal modes.
he Köppel−Domcke−Cederbaum (KDC) vibronic cou- overwhelmed in the moderate-dimensional case by its myriad − − T pling method1 4 is an often-employed approach5 16 for of possibilities. determining nonadiabatic spectral intensities, producing dis- Aiming at the second piece of the puzzle, in this work we crete line spectra. In its time-independent form, the KDC propose an algorithm for generating an optimal phonon algorithm uses a Lanczos procedure to diagonalize the molecular distribution of the final-state-biased normal modes. The
Hamiltonian in a direct product basis of harmonic nuclear algorithm provides a one parameter mapping N(M) where Nj vibrations and diabatic electronic states. A recently reported is the maximum number of phonons permitted in the jth mode. 17
Downloaded via JOHNS HOPKINS UNIV on August 20, 2020 at 14:59:38 (UTC). fi ne-grained parallel Lanczos solver is able to handle billions of M is described in eq 10, where it is used to optimize the total basis functions (if the eigenvectors are not demanded). overlap between the initial vibronic state and the final-state- However, the size of the vibronic coupling expansion grows
See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. biased basis. The comparatively small magnitude of N permits exponentially with respect to the number of vibrational modes, theLanczosroutineusedtodiagonalizethevibronic limiting its application to small molecules in the absence of Hamiltonian (eq 6) to employ explicitly orthogonalized Krylov compact vibrational bases. Compact vibrational bases are also vectors. demanded in variational methods for computing vibrational − In order to explain the construction of the vibronic basis we levels.18 21 begin with a brief review of the KDC approximation. The initial To determine a vibrational basis, normal modes and the vibronic state is given by number of phonons allowed in each mode are required. In typical cases, the spectral transition from a simple initial state initJ init Ψ (,rQ )= ϕψ ( Q ) (; rQ ) produces complicated final states, which are vibronically J (1) coupled levels whose electronic states can be coupled by conical − intersections.22 25 Some improvements have been made to Received: July 18, 2020 partially tackle the problem. A generating function technique26 Accepted: August 10, 2020 has eliminated the “Franck−Condon region as origin” Published: August 10, 2020 constraint, allowing final-state-biased normal modes to be − utilized.26 33 However, the phonon distribution has long been left to chemical intuition plus trial and error, which is
© XXXX American Chemical Society https://dx.doi.org/10.1021/acs.jpclett.0c02199 7245 J. Phys. Chem. Lett. 2020, 11, 7245−7252 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter where r denotes the electronic coordinates, Q is a set of nuclear 4). Diagonalization of eq 6 by a Lanczos routine is known to be mass-weighted normal coordinates to be defined below, ϕJ is the unstable, but still useful, when Nvib is sufficiently large that the nuclear wave function (usually the ground vibrational level, J =0, intermediate Krylov vectors cannot be stored and orthogonal- of the initial electronic state), and ψinit is the adiabatic electronic ized. However, Lanczos diagonalization performs well when it is wave function. The final vibronic state is expanded as feasible to orthogonalize the Krylov vectors. Limiting Nvib is thus
N state a key issue in this work and as we show below is precisely what final K d the proposed algorithm does. Ψ (,rQ )= ϕψ ( Q ) (; rQ ) K ∑ i i In the original formulation2 of the KDC approximation, the i=1 (2) ψinit fi normal modes of J (r; Q) were used to de ne (initial-state- state ψd 34,35 where N is the number of i , the quasidiabatic electronic biased) |n(Q)⟩. The motivation was to simplify the initial ψd J δ states. In photoelectron spectroscopy i has a continuum conditions (dm = mJ) required to describe short time 36−39 ϕK electron, which is suppressed. i , the vibrational function propagation. However, in the general case, the initial-state- ψd associated with i , is in turn expanded in a multimode biased basis is poorly positioned to describe the bound vibrational basis vibrational states of the final electronic state. Consequently, vib K N N must increase significantly to converge D .The fi ffi ϕKKi()QnQ=|⟩∑ D , () introduction of nal-state-biased bases improves the e ciency i n fi J n=0 of the spectral simulations but sacri ces the convenience of dm = δ 26 Nvib Nmode mJ. The previously mentioned generating function technique J Ki, enables convenient evaluation of d at a cost of only =|⟩≤<∑∏DnQnN(),0 n j j jj O(NvibNmode), making final-state-biased bases feasible. Several n=1 j=1 − (3) practical final-state-biased bases have been utilized.26 33 where |n (Q )⟩ denotes the n th harmonic oscillator function for To determine a vibrational basis, both |n(Q)⟩ and N are j j j fi mode j, Nj is the maximum number of phonons allowed in the jth required as shown in eq 3. The nal-state-biased normal modes mode (the phonon distribution), and Nvib is the size of the answer the choice of |n(Q)⟩, but N has long been left to vibrational basis chemical intuition plus trial and error. In fact, it is NP-hard to fi mode determine N:to nd some N producing a satisfactory spectrum N subject to Nvib(N) 7246 https://dx.doi.org/10.1021/acs.jpclett.0c02199 J. Phys.ji Chem. Lett. 2020,zy 11, 7245−7252 j z k { The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter Ω 1 N()LL= i 2 + ii ℏ i 2 (9b) Ω where i is the frequency corresponding to Qi. Since the multimode vibrational basis function is a direct product of single mode functions, the total effective volume of N is an Nmode- dimensional cuboid centered at the origin of Q. Given that the initial nuclear wave function is usually the ground vibrational level of the initial electronic state, e.g., photoelectron spectra are usually measured from cold anions, the harmonic ground state ϕ0(Q) dominates our consideration. The contour line (isoline) of ϕ0(Q), a product of Gaussian functions, is an ellipsoid centered at the initial state equilibrium geometry and, as one may infer, can be most efficiently covered by a circumscribed cuboid. This covering of the ellipsoid by the cuboid represents the overlap d. Certainly, covering a further isoline produces a larger d. A sketch to visualize this idea in the Nmode = 2 case can be found in Figure 1. Figure 2. (a) Cyclopentoxide anion and (b) cyclopentoxy radical ground state minima differ mainly in dihedral angles C3−C4−C5−C2 and C4−C5−C2−C1, corresponding to the lowest and the 2nd lowest frequency normal modes. Atom numbering used in text is also indicated. This is an ellipsoid equation with M adjusting the size of the ellipsoid. As shown in Figure 1, the greater M is, the larger the ellipsoid grows. d is thus proportional to M. The initial and the final states’ normal coordinate systems are related by q =+TQ b (11) Figure 1. Sketch of the continuum model for the optimal phonon fi T and b come from the definitions of q and Q; e.g., if in an distributions of nal-state-biased normal modes in a 2-dimensional fi case. Scan a batch of initial state contours (red ellipsoids labeled with M, arbitrary internal coordinate system the initial ( nal) state has ′ ″ M , M ), use eq 14 to obtain the outermost points along Q1 and Q2 (in equilibrium geometry Xi (Xf) and normal coordinate trans- min max this case Q1 and Q2 ), and determine corresponding distributions by formation Li (Lf), then eq 15 (blue cuboids circumscribing the red ellipsoids, labeled with N, −1 N′, N″). q =−LXXii() (12a) −1 Table 1. Representative Bond Lengths in Å, Bond Angles and Q =−LXXff() (12b) Dihedral Angles in Degree, for Cyclopentoxy Radical Ground −1 State Minimum (MIN), 1,22A Minimum Energy Crossing T = LLif (12c) (MEX), and Cyclopentoxide Anion Ground State Minimum −1 (AMIN) from Ref 45 bLX=−if() X i (12d) MIN MEX AMIN For a given M, we may map to an N(M)byfinding the C1−O 1.3795 1.3827 1.3321 circumscribed cuboid of the M contour ellipsoid. To determine fi C1−H1 1.0898 1.0914 1.1171 that cuboid, along each Qi direction we must nd the outermost O−C1−C2 105.4 112.2 112.8 point of the ellipsoid, which can be obtained by searching the O−C1−C3 111.1 109.8 112.8 ellipsoid along each Qi direction. In the Figure 1 example, the min max H1−C1−C2 111.8 112.6 108.2 leftmost point Q1 and the upmost point Q2 are where a H1−C1−C3 113.2 111.6 108.2 cuboid is tangent to an ellipsoid. More formally, this is a C4−C5−C2−C1 −18.1 −18.0 28.8 minimization (maximization) problem of Qi, subject to the C3−C4−C5−C2 37.6 37.6 0.0 constraint that the search point must stay on the ellipsoid. Mathematically, this is to minimize (maximize) the merit function Mathematically, let q and ω be the mass-weighted normal f ()Q = Q i (13a) modes and the frequencies of the initial state. Define an M subject to the constraint that Q lies on the ellipsoid contour T −12 Nmode cM()()()QTQbTQb≡+Σ+−=0 (13b) 2 2ωi 21†− M = ∑ ||=q q Σq We solve the constrained optimization problem eq 13 by ℏ i i=1 (10) optimizing the augmented Lagrangian40 7247 https://dx.doi.org/10.1021/acs.jpclett.0c02199 J. Phys. Chem. Lett. 2020, 11, 7245−7252 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter a Table 2. Phonon Distributions of Representative Final-State-Biased Bases Labeled with M Values ̈ 0.5 0.6 0.7 0.8 0.85 naive 11313141414 2 22628293031 2 322222 2 411111 2 511111 2 611112 2 711112 2 811112 2 911111 2 10 1 1 1 2 2 2 11 1 1 1 2 2 2 12 1 1 1 1 1 2 13 1 1 2 2 2 2 14 1 1 1 1 1 2 15 2 2 2 2 2 2 16 1 1 1 1 1 2 17 1 1 1 1 1 2 18 1 1 1 2 2 2 19 1 1 1 2 2 2 20 1 2 2 2 2 2 21 1 1 1 1 1 2 22 1 1 1 1 1 2 23 1 1 1 2 2 2 24 2 2 2 2 2 2 25 1 1 1 2 2 2 26 3 3 3 3 3 2 27 1 1 1 1 1 2 28 1 1 1 1 1 2 29 1 1 1 1 1 2 30 1 1 1 1 1 2 31 1 1 1 2 2 1 32 1 1 1 1 1 1 33 1 1 1 1 1 1 34 1 1 1 1 2 1 35 1 1 1 1 2 1 36 1 1 1 1 1 1 37 1 1 1 1 1 1 38 1 1 1 1 1 1 39 1 1 1 1 1 1 Nvib 8112 17 472 38 976 5 160 960 170 655 744 1 073 741 824 d 0.814 89 0.837 507 0.856 825 0.910 176 0.930 219 ∼0 a ̈ The basis origin is cyclopentoxy radical 1,22A minimum energy crossing. “Naive” denotes “2 phonons per mode except the C−H stretches” rather than an optimal distribution. 2 Lf()QQ=− ()λμ c () Q + c () Q (14) It is worthwhile to note that the d[N(M)] optimization can be reinterpreted: one can conveniently tune a final-state-biased where λ and μ are the augmented Lagrange multipliers. The basis by manipulating M, systematically approaching the λ μ initial values in our implementation are = 0 and = 0.5. Once complete basis limit in an optimal way. For example, one can min max the minimizer Qi (M) and the maximizer Qi (M)are diagonalize eq 6 using a small N(M) to simulate the entire determined, eq 9b gives Ni(M)as spectrum and obtain all important eigenstates and then repeat Ω 1 the calculation using a large N(M) focusing on the low levels to N()MQMQM= i max(2||||+ min (), max ()) establish convergence of the small N(M) result. This approach i ℏ ii 2 (15) can be used, for example, to include geometry-dependent spin− Ä É orbit coupling in a calculation. where [x] takesÅ the smallest integer no less than x. Thus,Ñ the − Å Ñ We apply our algorithm to cyclopentoxide anion (C H O ) minimum valueÅ of Ni is 1, which is consistent with eqÑ 9. The 5 9 ÇÅ ÖÑ photoelectron detachment to further illustrate our method. d(N) optimization problem is thus reduced to a simple − univariate one, namely, searching for the largest M subject to Figure 2 pictures C5H9O and the electron-detached species vib vib N [N(M)] < Naffordable. By analogy to 1-dimensional Gaussians, (cyclopentoxy radical, C5H9O) equilibrium geometries and where 2-sigma covers 95%, a satisfactory d[N(M)] should arise indicates the atom labeling. Table 1 provides the values of key − before M reaches 2, so M lies within a narrow range (0, 2). internal coordinates. C5H9O has Cs point group symmetry 7248 https://dx.doi.org/10.1021/acs.jpclett.0c02199 J. Phys. Chem. Lett. 2020, 11, 7245−7252 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter Figure 3. d values of final-state-biased bases as a function of (a) M and Figure 4. Based on the (a) M = 0.5 and (b) 0.6 bases (2nd and 3rd vib → (b) vibrational basis size N . Two series of bases are positioned at columns in Table 1), adding a phonon to a mode i (Ni:1 2) raises d cyclopentoxy radical 1,22A minimum energy crossing (MEX) and Δ ̈ by d. Activation of mode (a) 20 and (b) 13 produces the most gain. ground state minimum (MIN), respectively. “Naive” denotes “2 phonons per mode except the C−H stretches” rather than an optimal ffi distribution. electron a nity calculated by the MRCISD and the CISD electronic energies and harmonic frequencies is 1.2 eV. The experimental spectrum55 is measured from cryogenic − C5H9O photoionized at 532 nm (and 355 nm, omitted due to while C5H9OhasC1 symmetry. This symmetry breaking no new feature). The independence of photoionization to corresponds to huge 40° distortions in dihedral angles C3− wavelength validates the Franck−Condon approximation. The C4−C5−C2 and C4−C5−C2−C1. The two lowest states of broad peak observed in experiment supports a large geometry − ff − C5H9O have a low-lying conical intersection producing a Jahn di erence between C5H9O and C5H9O. The measured Teller type interaction41 that couples the corresponding adiabatic electron affinity is 1.5 eV. potential energy surfaces. The huge 40° distortions in dihedral angles C3−C4−C5−C2 − − − ffi The C5H9O conical intersection necessitates the use of a 2- and C4 C5 C2 C1 increase the di culty in using the KDC state quasidiabatic Hamiltonian (Hd) to describe the electronic approximation to describe vibronic coupling. First, it is hopeless structure. Hd was constructed in a previous work42 using to manually determine the phonon distribution. In a previous multireference configuration interaction with single and double work on propyne,33 the overlap between the harmonic excitations (MRCISD) wave functions based on 6 frozen core vibrational ground states of the neutral and the cation is already − orbitals, 15 doubly occupied orbitals, and a 5-electron 3-orbital 0.7, while that of C5H9O plunges to 0.000 024 (recall from eq 8 active space. An atomic orbital basis set which is cc-pVTZ43 for that d → 1 is a necessary condition of the basis adequacy). As a the oxygen and the carbons and cc-pVDZ43 for the hydrogens consequence, for propyne it is tolerable to determine the was used. The MRCISD expansion is composed of 19 302 445 phonon distribution by trial and error, while that becomes fi − con guration state functions (CSFs). For the closed-shell anion hopeless for C5H9O . Second, many phonons must be placed in − − − − − C5H9O the above-described MRCISD was replaced by a single the normal modes corresponding to C3 C4 C5 C2 and C4 reference CISD expansion with 6 frozen core orbitals and 18 C5−C2−C1, leading to large vibronic Hamiltonian matrix doubly occupied orbitals, with the atomic orbital basis set for elements, deteriorating the fragile numerical stability of the oxygen extended to aug-cc-pVTZ.44 The CISD expansion is Lanczos algorithm. This is the case encountered in this work: composed of 7 634 278 CSFs. All electronic structure data used Even for the smallest basis tested (Nvib ∼ 104), the zero-point in this work were reported previously45 and employed the energy can be in error by 7 eV without orthogonalizing the − COLUMBUS46 54 suite of programs. The MRCISD evinces a Krylov vectors. This orthogonalization is feasible only with the 2 vib iter 1,2 A minimum energy crossing (MEX) structure for C5H9O compact basis, since there are additional O(N N ) storage that deviates slightly from its ground state minimum (MIN) and and O(Nvib[Niter]2) computational requirements in the is only 0.064 eV higher in energy (see Table 1). The adiabatic orthogonalized Lanczos diagonalization, where Niter is the 7249 https://dx.doi.org/10.1021/acs.jpclett.0c02199 J. Phys. Chem. Lett. 2020, 11, 7245−7252 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter M, as expected by our continuum model: the covering of the ellipsoid by the cuboid is equivalent to the overlap d, and covering a larger isoline (M) produces a greater d. The dash lines are the linear regressions, whose coefficients of determination are both 96%, and extrapolations predict that for MEX (MIN) d → 1 will be reached when M → 1.08 (0.90). Exact d values cannot be computed for such M due to the overwhelming Nvib. Figure 3b shows that, despite the large molecular size, the optimal phonon distribution can give small but adequate bases. The details of phonon distributions for the MEX bases can be ̈ found in Table 2. The performance of the smallest naive distribution (2 phonons per mode except the C−H stretches) is also reported Figure 3b. Although its size exceeds one billion, the resulting d value is less than 0.05. On the contrary, our optimized distributions routinely give d > 0.7 ≫ 0.05 requiring only Nvib ∼ 104. From Figure 4 and Table 2 we illustrate that our continuum model always finds the optimum mode(s) to improve an existing d[N(M)]. From Table 2, we see that the largest change from M = 0.5 to 0.6 (M: 0.5 → 0.6) is the activation of the single, → previously frozen, mode 20 (N20:1 2). Similarly the result for → → M: 0.6 0.7 is N13:1 2. Figure 4a,b, respectively, shows that, not surprisingly, placing a phonon in mode 20 (13) produces the largest increase in d on any single replacement as our continuum model predicts. In the general case, when more than one mode is excited manual trial and error selection of groups of modes is no longer feasible, but one can always rely on our continuum model. We demonstrate the fast convergence of our final-state-biased bases in Figure 5. The MEX basis is chosen since it is found in Figure 3 to perform better when Nvib is small. Figure 5a reports Figure 5. (a) Intensity of the max intensity line as a function of the Nvib dependence of the intensity of the maximum intensity vibrational basis size Nvib. (b) Energies of cyclopentoxy radical vibronic line (MIL). This figure shows that the MIL converges as a ground state, 1st excited state, and max intensity state. The zero of vib energy is taken as that of cyclopentoxy radical ground state minimum. function of N . Particularly gratifying is the small change in the MIL with large change in Nvib for Nvib large. Similar convergent behavior is reported in Figure 5b for the energies of the ground state (the zero-point energy), the first excited state, and the MIL. These observations are quantified by the fits to the intensity and the energies provided by the dashed lines in Figure 5.A complete basis set limit extrapolation of any observable O is given by vib a O()N =+O∞ Nvib (16) where a and O∞ are extrapolation parameters. From the analyses Nvib = 38 976 (M = 0.7, d = 86%) is seen to well converge the MIL, while the low-lying states require Nvib = 5 160 960 (M = 0.8, d = 91%) to converge reasonably. The faster convergence of the MIL is as anticipated, since our continuum model is designed to bias the MIL. Given Nvib = 5 160 960 (M = 0.8, d = 91%) has proven to be Figure 6. Simulated photoelectron spectrum of cyclopentoxide anion, in comparison to the supersonic expansion cryogenic experiment from converged in Figure 5, we simulate the photoelectron spectrum ref 55. A Gaussian function with 0.01 eV width is adopted for with this basis, and the result is presented in Figure 6. A Gaussian convolution. function with 0.01 eV width is adopted for convolution. For comparison, the simulated and measured intensity maxima are made to agree, accounting principally for the difference in the number of iterations. Given our resources (1 TB hard disk and calculated and the measured adiabatic electron affinities. The vib 7 24 CPUs), N <10 is the most we can afford to simulate the simulation well reproduces the experiment: the peak heights entire spectrum and obtain all important eigenstates. match the experiment between 0.3 and 1.2 eV; the convolution Figure 3 demonstrates two series of final-state-biased bases overlays the experiment between 0 and 0.2 eV. The convolution positioned at C5H9O MEX and MIN. The normal modes are is not as broad as the experiment between 0.2 and 1.2 eV. dd+ defined by diagonalizing ∇∇ HH11 22 and the adiabatic To sum up, we have reported a continuum model enabling ()2 automatic generation of compact multimode vibrational bases ground state Hessian, respectively. Figure 3a shows that d ∝ for spectral simulations of moderate-sized molecules. The 7250 https://dx.doi.org/10.1021/acs.jpclett.0c02199 J. Phys. Chem. Lett. 2020, 11, 7245−7252 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter photoelectron spectrum of cyclopentoxide has been determined (11) Ichino, T.; Wren, S. W.; Vogelhuber, K. M.; Gianola, A. J.; using this algorithm and compared with experiment. The Lineberger, W. C.; Stanton, J. F. The Vibronic Level Structure of the compact representation can be adapted to incorporate the Cyclopentadienyl Radical. J. Chem. Phys. 2008, 129, 084310. geometry-dependent spin−orbit interaction into the non- (12) Stanton, J. F. On the Vibronic Level Structure in the NO3 relativistic eigenstate representation. In the future these tools Radical. I. The Ground Electronic State. J. Chem. Phys. 2007, 126, will be applied to analyze fine-structure quenching in cyclo- 134309. (13) Stanton, J. F. On the Vibronic Level Structure in the NO3 pentoxy vibronic wave functions. Radical: II. Adiabatic Calculation of the Infrared Spectrum. Mol. Phys. 2009, 107, 1059−1075. ■ AUTHOR INFORMATION (14) Stanton, J. F.; Okumura, M. On the Vibronic Level Structure in Corresponding Authors the NO3 Radical : Part III. Observation of Intensity Borrowing via Yifan Shen − Department of Chemistry, Johns Hopkins University, Ground State Mixing. Phys. Chem. Chem. Phys. 2009, 11, 4742. Baltimore, Maryland 21218, United States; (15) Weichman, M. L.; Cheng, L.; Kim, J. B.; Stanton, J. F.; Neumark, orcid.org/0000- D. M. Low-Lying Vibronic Level Structure of the Ground State of the 0003-0590-444X; Email: [email protected] Methoxy Radical: Slow Electron Velocity-Map Imaging (SEVI) Spectra David R. Yarkony − Department of Chemistry, Johns Hopkins ̈ University, Baltimore, Maryland 21218, United States; and Koppel-Domcke-Cederbaum (KDC) Vibronic Hamiltonian Calculations. J. Chem. Phys. 2017, 146, 224309. orcid.org/0000-0002-5446-1350; Email: [email protected] (16) Rabidoux, S. M.; Cave, R. J.; Stanton, J. F. Nonadiabatic Investigation of the Electronic Spectroscopy of trans-1,3-Butadiene. J. Complete contact information is available at: − https://pubs.acs.org/10.1021/acs.jpclett.0c02199 Phys. Chem. A 2019, 123, 3255 3271. (17) Schuurman, M. S.; Young, R. A.; Yarkony, D. R. On the Multimode Quadratic Vibronic Coupling Problem: An Open-Ended Notes Solution Using a Parallel Lanczos Algorithm. Chem. Phys. 2008, 347, fi The authors declare no competing nancial interest. 57−64. (18) Light, J. C.; Hamilton, I. P.; Lill, J. V. Generalized Discrete ■ ACKNOWLEDGMENTS Variable Approximation in Quantum Mechanics. J. Chem. Phys. 1985, − This work was supported by National Science Foundation grant 82, 1400 1409. (19) Bowman, J. M.; Carrington, T.; Meyer, H.-D. Variational (CHE-1663692) to D.R.Y. The authors are grateful for a grant of Quantum Approaches for Computing Vibrational Energies of computer time on the Maryland Advanced Research Computing Polyatomic Molecules. Mol. Phys. 2008, 106, 2145−2182. Center (MARCC) at Johns Hopkins University. The authors are (20) Carter, S.; Handy, N. C. A Variational Method for the grateful to Professor Lan Cheng (Johns Hopkins University) for Calculation of Vibrational Levels of Any Triatomic Molecule. Mol. sharing hard disk and Professor Robert Continetti (University of Phys. 1982, 47, 1445−1455. California, San Diego) for helpful discussion on the details of his (21) Carter, S.; Handy, N. C. The Variational Method for the experiment. 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